0213_0231_0312_0321_2130
Counting sequence:
1, 1, 2, 6, 19, 56, 159, 444, 1224, 3340, 9043, 24334, 65161, 173800, 462086, 1225354, 3242419, 8564636, 22589851, 59510084, 156613468, 411814816, 1082105139, 2841721986, 7458988085, 19570370960, 51329603562, 134589324526, 352815268307, 924687755744, 2423081434423, 6348600316812, 16631638329936, 43565985736340, 114109704775059, 298858819194710, 782677805084161, 2049640086503928, 5367269125925806, 14054431686544946, 36801020214698099, 96359644191025668, 252302207220602995, 660600561476220564, 1729617660452409508, 4528513081231933672, 11856496489950680179, 31042244385279035018, 81273033320555206893, 212783023792431438368, 557089642485487741778, 1458515909176197836118, 3818524264283482717587, 9997202846583752852232, 26173406206798946959439, 68523725815715807217756, 179399337287063415612760, 469677740070235492578716, 1229641501015067887231443, 3219263565204532760250654, 8428186253082420628748793, 22065376929101934421560328, 57768124806571357818196438, 151239395093791979632785754, 395950937416223467575238963, 1036615351310064171265351660, 2713899382427520383165413835, 7105092204741018538615163620, 18601397983487764101621045100, 48699147515020282841074505648, 127496145508937624131639813555, 333789511658213897842860586514, 873872880527844695049799782949, 2287830212996965978322291777968, 5989620147252766130237597233722, 15681035497402898818684401125694, 41053497965310708929419161563987, 107479484028028498067100716090288, 281385010646414891873232152957415, 736675672586851169358897630703660, 1928642282094907869913592147520992, 5049251780187051054403490794844708, 13219114396120237513146406090905619, 34608094358462415178444910594024550, 90605175186333694023028303906458225, 237207445552326031112169483409823000, 621017193124507881449948294008108926, 1625844203635991263511451672199990274, 4256515571764840573853488752047307763, 11143702851275143269723646790537922900, 29174593731108608508940492306343927203, 76380079994128095469112993531504237108, 199965649895047117134103447300785530804, 523516877727604153678230307082863314936, 1370585001013024552602668554774836711667, 3588238164405759360769642477908328162074, 9394129578429424140731026078995521418205, 24594150761058113492175051239996015677504, 64388323124190407053937225723494766126242, 168570819536629259716947589295538137368742, 441324137526105276622279084374137134899987
Generating function in Maple syntax:
(x^6+x^5-3*x^4-4*x^2+4*x-1)/(x^2-3*x+1)/(x^3+2*x-1)
Generating function in latex syntax:
\frac{x^{6}+x^{5}-3 x^{4}-4 x^{2}+4 x -1}{\left(x^{2}-3 x +1\right) \left(x^{3}+2 x -1\right)}
Generating function in sympy syntax:
(x**6 + x**5 - 3*x**4 - 4*x**2 + 4*x - 1)/((x**2 - 3*x + 1)*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(x^3+2*x-1)*F(x)+x^6+x^5-3*x^4-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(x^{3}+2 x -1\right) F \! \left(x \right)+x^{6}+x^{5}-3 x^{4}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
341/295*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^5-3*_Z^4+3*_Z^3-7*_Z^2+5*_Z-1))-959/295*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^5-3*_Z^4+3*_Z^3-7*_Z^2+5*_Z-1))+702/295*Sum(_alpha^(1-n),_alpha = RootOf(_Z^5-3*_Z^4+3*_Z^3-7*_Z^2+5*_Z-1))-4783/590*Sum(_alpha^(-n),_alpha = RootOf(_Z^5-3*_Z^4+3*_Z^3-7*_Z^2+5*_Z-1))+1873/590*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5-3*_Z^4+3*_Z^3-7*_Z^2+5*_Z-1))+piecewise(n = 0,4,n = 1,1,0)
Explicit closed form in latex syntax:
\frac{341 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}-7 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{295}-\frac{959 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}-7 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{295}+\frac{702 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}-7 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{295}-\frac{4783 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}-7 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{590}+\frac{1873 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}-7 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{590}+\left(\left\{\begin{array}{cc}4 & n =0 \\ 1 & n =1 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 159
a(n+5) = a(n)-3*a(n+1)+3*a(n+2)-7*a(n+3)+5*a(n+4), n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 56
a \! \left(6\right) = 159
a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+3 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20204/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[10,x]*F[4,x]
F[4,x] = F[11,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = x
F[11,x] = F[12,x]+F[22,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[15,x]+F[5,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[12,x]
F[22,x] = F[23,x]+F[24,x]+F[38,x]
F[23,x] = 0
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[26,x]+F[32,x]
F[26,x] = F[27,x]+F[6,x]
F[27,x] = F[23,x]+F[28,x]+F[30,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[26,x]
F[30,x] = F[10,x]*F[31,x]
F[31,x] = F[9,x]
F[32,x] = F[22,x]+F[33,x]
F[33,x] = 2*F[23,x]+F[34,x]+F[36,x]
F[34,x] = F[10,x]*F[35,x]
F[35,x] = F[32,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[20,x]
F[38,x] = F[10,x]*F[39,x]
F[39,x] = F[11,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[21,x]+F[23,x]+F[42,x]
F[42,x] = F[10,x]*F[43,x]
F[43,x] = F[44,x]+F[49,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[23,x]+F[46,x]+F[48,x]
F[46,x] = F[10,x]*F[47,x]
F[47,x] = F[44,x]
F[48,x] = x^2
F[49,x] = F[20,x]+F[50,x]
F[50,x] = 2*F[23,x]+F[51,x]+F[53,x]
F[51,x] = F[10,x]*F[52,x]
F[52,x] = F[49,x]
F[53,x] = F[10,x]*F[20,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{22}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{5}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{38}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{32}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{6}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{9}\! \left(x \right)
F_{32}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{32}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{21}\! \left(x \right)+F_{23}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = x^{2}
F_{49}\! \left(x \right) = F_{20}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right) F_{20}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_10(x)*F_4(x))
Eq(F_4(x), F_11(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_22(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_15(x) + F_5(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_12(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_38(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_32(x))
Eq(F_26(x), F_27(x) + F_6(x))
Eq(F_27(x), F_23(x) + F_28(x) + F_30(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_26(x))
Eq(F_30(x), F_10(x)*F_31(x))
Eq(F_31(x), F_9(x))
Eq(F_32(x), F_22(x) + F_33(x))
Eq(F_33(x), 2*F_23(x) + F_34(x) + F_36(x))
Eq(F_34(x), F_10(x)*F_35(x))
Eq(F_35(x), F_32(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_20(x))
Eq(F_38(x), F_10(x)*F_39(x))
Eq(F_39(x), F_11(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_21(x) + F_23(x) + F_42(x))
Eq(F_42(x), F_10(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_49(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_23(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_10(x)*F_47(x))
Eq(F_47(x), F_44(x))
Eq(F_48(x), x**2)
Eq(F_49(x), F_20(x) + F_50(x))
Eq(F_50(x), 2*F_23(x) + F_51(x) + F_53(x))
Eq(F_51(x), F_10(x)*F_52(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_10(x)*F_20(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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